Magnetic Bearings: Purposes and Attributes.
The purpose of a magnetic bearing is to provide a force between two major bearing members without contact occurring. This force is subsequently referred to as the bearing force. Consistent with the normal definition of any bearing, a magnetic bearing allows free motion in one or more senses whilst providing the capability for exerting bearing forces in at least one other sense. Most magnetic bearings are employed in rotating machines to separate the rotor and the stator. Magnetic bearings have the advantages of very low energy loss rate (given proper design), no contact between parts giving potentially very long life, and the ability to withstand relatively high temperatures.
Magnetic bearings may be active or passive. Active magnetic bearings sense the relative position of the two major bearing members. They then adjust the electric currents in coils such that the net force between the two major bearing members has the appropriate magnitude and direction. Passive magnetic bearings usually involve magnetic fields from permanent magnets but they may alternatively be constructed using coils of conductor to provide the magnetomotive force (MMF). The electric currents flowing in these coils are not, however, a strong function of the relative position of the two bearing members. Passive magnetic bearings often operate on the basis of repulsion of like poles.
In a simple view, active magnetic bearings may be arbitrarily stiff in the sense that the smallest amount of relative movement between the two major bearing members can be made to cause a finite amount of force. There are obviously limitations to this associated with the ability to sense extremely small motions and the need for the closed-loop control system to be stable. However, it is broadly accepted that active bearings are generally orders of magnitude more stiff than their passive equivalents. Stiffness of a bearing is extremely important for acceptable dynamic properties and in order that the relative position of the two major bearing members is insensitive to the externally-applied load existing between them.
Another extremely important attribute of any bearing is reliability. Active magnetic bearings are complicated systems involving sensing, control, and power-currents. As such, there are very many possible modes of failure other than straightforward mechanical breakage. By contrast, passive bearings tend to be extremely robust and reliable with very few modes of possible failure other than mechanical breakage.
A key attribute of any magnetic bearing is size. A second related and equally important attribute is total weight. It is generally accepted that for a given force rating, a radial magnetic bearing is many times larger than its rolling-element counterpart.
Elementary Bearing Regions and the Central Surface.
Consider bearings which comprise two major bearing members between which some free relative motion is to be provided. In many cases, at least one of the free relative motions is a rotation. The rotation of any physical body of a scale above atomic scale involves translation of the particles at the surface of that body. To provide a bearing which can offer forces to resist relative motions of the two major bearing members in some senses and yet allow free rotation about some axis, it is both necessary and sufficient to provide regions within the bearing where translation is opposed along at least one axis and free along at least one other axis. Such regions are referred to as elementary bearing regions.
The conventional ball bearing is useful for illustration. This bearing contains a finite number of elementary bearing regions—one per ball—where relative translation of the inner race and the outer race is stiffly resisted along one direction and where translation of the inner race and outer race is free in the other two directions. The stiff direction for each individual ball at a given instant is along the (ball) diameter between contacts. This simple conceptual model obviously ignores friction and viscous shear forces at the contacts. FIG. 1 illustrates the elementary bearing region of a ball bearing.
The collective action of all of these elementary bearing regions in the case of a ball bearing results in a bearing which provides free rotation about one axis but reacts against all net translations between the two major bearing members and (for an angular contact ball bearing) against the rotations about the other two orthogonal axes of rotation.
A similar view can be taken of a cylindrical roller bearing. Each roller can produce very stiff opposition to relative translations of the inner and outer race in one direction. It will allow very free movement in the direction of rolling. It provides some resistance to relative movement of the inner and outer races in the axial direction although this resistance is not usually used. For cylindrical roller bearings, we can consider that there is one elementary bearing region for each individual roller. For roller bearings having conical rollers, consider each roller to comprise a large number of disc-like slices and the elementary bearing regions are revealed. FIG. 2 illustrates an elementary bearing region from a conical roller bearing.
It is straightforward to extend this view of all bearings which accommodate rotation to hydrostatic and hydrodynamic bearings. In the case of hydrostatic bearings, the elementary bearing regions can be regarded as the individual locations where pressurised fluid is fed into the cavity between the two major bearing members. FIG. 3 depicts an elementary bearing region from a hydrostatic bearing, showing, superimposed, a pressure distribution over such a location. In the case of hydrodynamic bearings, the lubricant interlayer between the two major bearing members can be decomposed into patches each of which exerts some force to maintain a distance between the two major bearing members. FIG. 4 illustrates one such patch, and the direction of relative motion between the bearing members.
Following the above logic, all bearings can be decomposed into sets of elementary bearing regions having at least one direction of comparatively free relative translation and at least one direction in which translation is (or can be) strongly opposed.
In each of the above examples of bearings, the elementary bearing regions include a portion of the surface of each of the two major bearing members. Between these two surfaces, there is a central surface. This is any smooth surface such that the action of the bearing region in providing a direction of free translation can be considered to be equivalent to sliding of one side of this central surface relative to the other. The term central surface is used regularly throughout the remainder of this document.
In most cases, the elementary bearing regions are only (or predominantly) used to provide free translation in one direction. This direction is in the plane of the central surface. Thus, it is possible to establish an axis set of principal directions for an elementary bearing region according to FIG. 5 in which the three axes are:    (1) The axis of (predominant) free relative translation. For obvious practical reasons, the free relative translation is similar to a discrete pure shearing action at the central surface. This direction is arbitrarily labelled x in FIG. 5.    (2) The axis normal to the central surface. This direction is arbitrarily labelled z in FIG. 5.    (3) The remaining orthogonal axis, labelled direction y in FIG. 5.
In all of the above cases, the force acting between the two surfaces of the two major bearing members is predominantly along the normal to the central surface i.e. along the z direction of FIG. 5.
No practical bearing at scales above atomic scales is truly lossless. There is some rolling resistance in ball and roller bearings. There is some viscous drag in the bearing fluids in hydrostatic and hydrodynamic bearings. There are eddy-current losses and hysteresis losses in magnetic bearings. Thus, in all cases, there is invariably some component of force acting to oppose the relative translation of the two surfaces in the “free” direction, x.
Magnetic Stresses in Magnetic Bearings
Many existing designs of magnetic bearings rely squarely on the fact that where magnetic flux is caused to pass through air, there is effectively a tensile Maxwell stress in the air in the direction of the lines of magnetic flux. Most, if not all, active magnetic bearings currently available operate directly on the basis of this tensile stress.
FIG. 6 illustrates the action of the tensile Maxwell stress in probably the simplest instance where a horse-shoe shaped permanent magnet drives a magnetic field through itself, an airgap (twice) and some second body. Because the lines of magnetic flux in this case are predominantly normal to the faces of the horse-shoe magnet and to the surface of the second body, it is possible to approximate the net force generated at each of the two airgap-crossings by a simple formula. These two discrete forces can then be combined using elementary trigonometry to produce an expression for the total resultant attractive force between the magnet and the second body.
The oldest designs of active magnetic bearing comprise separately-energised horse-shoe shaped electromagnets arranged about the circumference of an airgap with a solid (or hollow) cylindrical rotor in the centre. Each horse-shoe electromagnet has its own complete magnetic circuit and there is very little interaction between distinct electromagnets. In normal operation, each electromagnet has a bias field such that there is always some magnetic flux through the horse-shoe electromagnet. The bias field is sometimes provided by a DC component of current in the electromagnet but it can be provided by a permanent magnet in the magnetic circuit. The forces produced by the bias fields generally sum to near zero. Then by introducing a relatively small amount of (additional) current in one horse-shoe electromagnet and the negative of this (additional) current in the horse-shoe electromagnet diametrically opposite, a net transverse force is created between the bearing stator and the bearing rotor.
Some more modern designs of magnetic bearing utilise stator shapes which are akin to the stators of switched-reluctance machines in that there are inwardly-protruding stator poles mounted onto a continuous cylinder of back-iron. There may be coils on individual stator poles or coils may link two or more poles. Alternatively coils may be formed around the back of core following the old Gram-ring winding method which was common in electrical machines some years ago. Permanent magnets may be provided in the stator poles or in the cylinder of back-iron to create the bias field. The relationship between individual currents in coils (or phases) and the quantity of magnetic flux passing through the individual stator poles is more complex in these cases than it is for the simple arrangement of multiple independent horse-shoe electromagnets. However the basic principle of operation is the same: attractive force per pole is (roughly) proportional to the square of total flux through the pole-face.
Most magneto-mechanical devices are fundamentally limited by flux density. It is very rare for flux densities in any iron-containing machine to rise above 2 Tesla anywhere in the iron because of saturation. (The word iron is used here to encompass any ferromagnetic material). Maximum flux density in a ferro-magnetic material is a key parameter in choosing such a material for an application but it is not the only one. Mechanical strength, stiffness, resistivity (for eddy-current losses) and low magnetic hysteresis effects are other properties that the designer must keep in mind when selecting a material for use in a magneto-mechanical device. Of course, there is ultimately no maximum magnetic flux density in iron or any other material but the (incremental) relative permeability for iron can fall from over 1000 at low flux levels to not much above 1 at flux levels over 2 Tesla.
Magnetic flux densities in the iron of an iron-carrying magneto-mechanical device are invariably higher than those in the airgaps where the magnetic flux is effective in generating force. The term airgap is used in this context to mean a region of space that may or may not be filled by a non-magnetic fluid. This usage is consistent with the interpretation of the term in the context of electrical machines. Most usually, the gap between relatively movable parts of the device is occupied by air.
Given that airgap flux density is limited, it follows that the Maxwell stresses achievable are also limited in magnitude. The net force or torque acting through an airgap can be computed by choosing any surface through that airgap and integrating the magnetic stresses over that surface. If this is done, an average effective airgap stress can be derived as the total force divided by the total airgap area or the total torque divided by the total first-moment of airgap area about the axis of rotation. The average airgap stress is limited to about 0.4 MPa.
In the context of the design of any magnetic bearing, a key requirement is to be able to develop a certain nominal force capable of resisting motion in one direction. Given that the effective airgap stress in any magneto-mechanical device is inherently limited by saturation of iron, it follows that there is a minimum operative area of airgap for a given rated load. One route taken by designers of magnetic bearings is to use relatively large flat bearing surface areas through which magnetic flux passes. Another route taken is to use relatively large-diameter/long bearing surfaces so that the requisite airgap area can be achieved in a finite length of shaft.
For a given magnetic flux density, B, in the airgap, the tensile Maxwell stress in the direction, “r”, of the lines of flux is given by:
      σ    rr    =            B      2              2      ⁢              μ        0            
A fact that is much neglected in the design of magnetic bearings is that in the two directions, “s” and “t”, perpendicular to r there is effectively a compressive stress given by:
      σ    ss    =            -                        B          2                          2          ⁢                      μ            0                                =          σ      tt      
FIG. 7a shows a set of magnetic flux lines in a plane of constant t. The square box drawn in FIG. 7a can be considered to have tension, σrr, acting on two opposite faces and compression (negative tension), σss, acting on the other two opposite faces. FIG. 7b shows the same set of magnetic flux lines in the same plane of constant t. A square box of the same size as that in FIG. 7a is drawn here also but the orientation of this square box is at 45° to the orientation of the box in FIG. 7a. In this figure, axes “u” and “v” are defined to occur at 45° angles to the direction of the magnetic flux. On the sides of this box, it is found that effectively a pure shear stress is acting with no component of normal stress. The magnitude of this pure shear stress “τUV” (in FIG. 7b) is identical to the magnitude of the normal stresses on the sides of the box in FIG. 7a.
      τ    ss    =            B      2              2      ⁢              μ        0            
Returning to the discussion of elementary bearing regions, consider that lines of magnetic flux are passing between the two bounding surfaces of the elementary bearing region in FIG. 5 such that each flux line is (at least approximately) perpendicular to the x direction (the direction in which free relative motion of the two bounding surfaces is desired). Provided that this condition is satisfied, there will be component of force between the two major components in the x direction. If these lines of flux are parallel to the z direction (normal to the central surface), then the force between the two bounding surfaces will equal to the stress times the area, i.e. B2A/2 μ0 where B is the flux density and A is the area of the central surface.
If, as indicated in FIG. 8, the flux lines are all perpendicular to x and they lie at an angle α to the normal, Z, then there will be components of force between the two bounding surfaces of the elementary bearing region in directions y and z, given by:
      F    y    =                                          B            2                    ⁢          A                          2          ⁢                      μ            0                              ⁢              sin        ⁡                  (                      2            ⁢            α                    )                    ⁢                          ⁢              F        z              =                                        B            2                    ⁢          A                          2          ⁢                      μ            0                              ⁢              cos        ⁡                  (                      2            ⁢            α                    )                    
In FIG. 8, positive Fy acts to pull the upper bounding surface in the −y direction and it acts to pull the lower bounding surface in the +y direction. Positive Fz acts to pull the upper bounding surface in the −z direction and it acts to pull the lower bounding surface in the +z direction.